Well, it looks as though Hanman is a real possibility. And possibly Churcham is de Eaden (the children were baptised in Churcham).

But I am not sure how to chase this further - I cannot find Alice's or Jonathan's baptism - or even burials. If they were married in St. Peter's 'by invitation' could they be buried there?. They were presumably born c 1700 at the latest, more likely 1690ish. This makes Henry Hanman and Anna Ball the only likely candidates for Alice's parents.

The Hares seem to have large gaps...